Question 1
Let $n \ge 3$ be an integer number. We say that a matrix $A \in \mathcal{M}_n(\mathbb{C})$ has the property $(\mathcal{P})$ if $\det(A + X_{ij}) = \det(A + X_{ji})$, for any $i, j \in \{1, 2, \dots, n\}$, where $X_{ij} \in \mathcal{M}_n(\mathbb{C})$ is the matrix with $1$ at the position $(i, j)$ and $0$ elsewhere.
a) Assume a matrix $A \in \mathcal{M}_n(\mathbb{C})$ with the property $(\mathcal{P})$, such that $\det(A) \neq 0$. Prove that $A = A^T$.
b) Give an example of a matrix $A \in \mathcal{M}_n(\mathbb{C})$ with the property $(\mathcal{P})$, but $A \neq A^T$.
a) Assume a matrix $A \in \mathcal{M}_n(\mathbb{C})$ with the property $(\mathcal{P})$, such that $\det(A) \neq 0$. Prove that $A = A^T$.
b) Give an example of a matrix $A \in \mathcal{M}_n(\mathbb{C})$ with the property $(\mathcal{P})$, but $A \neq A^T$.